Generating the pattern should be relatively easy using an extension.
-- Draw a rectangle the size you want
-- Path menu > Object to Path
-- Extensions menu > Generate from Path > Voronoi Pattern (Click the Live Preview button, and you can try out different values in a live way. When you have a pattern that you like, click the Apply button.)
-- Object menu > Pattern > Pattern to Object
If you deselect everything at this point, and do Ctrl A (select all) you'll see that you now have the original rectangle, and the pattern (which was just turned to a path). They're right on top of each other , so you might need to use the status bar to confirm that both are selected.
-- Object menu > Fill and Stroke > Stroke Style tab > Width (Make the lines the width you want.)
-- Path menu > Stroke to Path
-- Path menu > Union
That should make the lines mostly with equal widths. But around the intersections, there could be some less than consistent widths. So you might want to use the Node tool to fine tune those areas a little bit.
Are you already familiar with editing nodes, or do you need some tips?